Deviation risk measure

In financial mathematics, a deviation risk measure is a function to quantify financial risk (and not necessarily downside risk) in a different method than a general risk measure. Deviation risk measures generalize the concept of standard deviation.

A function ${\displaystyle D:{\mathcal {L}}^{2}\to [0,+\infty ]}$, where ${\displaystyle {\mathcal {L}}^{2}}$ is the L2 space of random portfolio returns, is a deviation risk measure if

There is a one-to-one relationship between a deviation risk measure D and an expectation-bounded risk measure R where for any ${\displaystyle X\in {\mathcal {L}}^{2}}$

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